   Chapter 11.10, Problem 10E

Chapter
Section
Textbook Problem

Use the definition of a Taylor series to find the first four nonzero terms of the series for f ( x ) centered at the given values of a. f ( x ) = cos 2 x ,     a = 0

To determine

To find:

The first four nonzero terms of Taylor series

Explanation

1) Concept:

Taylor series of the function   f at   a is

fx=n=0fnan!(x-a)n=fa+f'a1!(x-a)+f''a2!(x-a)2+f'''a3!(x-a)3

2) Given:

fx=cos2x,   a=0

3) Calculation:

As   a=0, Taylor series is given by:

fx=n=03fn0n!(x)n=f0+f'01!x+f''02!x2+f'''03!x3+

Let’s find the coefficients of this series.

fx=cos2x

So,

f0=cos20=1

Differentiate   f(x) with respect to x

f'x=2cosx·(-sinx)

f'x=-sin2x

So,

f0=-sin0=0

Now, differentiate   f'(x) with respect to x to get f''x

f''x=-cos2x·2

f''x=-2cos2x

So,

f''0=-2cos2x=-2

Now, using the product rule, differentiate f''x with respect to x to get f'''x

f'''x=-2·-sin2x·2

f'''x=4sin2x

So,

f'''0=4sin0=0

As the two terms so far of this series are zero, let’s find the next terms

Now, using the product rule, differentiate f'''x with respect to x to get f''''x

f''''x=4·cos2x·2

f''''</

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